# When the SuperBall® was introduced in the 1960’s, kids across the United States were amazed that these hard rubber balls could bounce to 90% of the height from which they were dropped. Is this...

When the SuperBall® was introduced in the 1960’s, kids across the United States were amazed that these hard rubber balls could bounce to 90% of the height from which they were dropped.

- Is this problem an example of a geometric series or an arithmetic series? Support your answer mathematically by applying the concepts from this unit.
- If a SuperBall® is dropped from a height of 2m, how far does it travel by the time it hits the ground for the tenth time? (Hint: The ball goes down to the first bounce, then up and down thereafter.)

You borrowed $5,000 from your parents to purchase a used car. You have agreed to make payments of $250 per month plus an additional 1% interest on the unpaid balance of the loan.

- Is this problem an example of a geometric series or an arithmetic series? Support your answer mathematically by applying the concepts from this unit.
- Find the first year’s monthly payments that you will make and the unpaid balance after each month.
- Find the total amount of interest paid over the term of the loan.

- Use mathematical induction to prove the property for all positive integers
*n*. [a^n]^5=a^5n - Use mathematical induction to prove the property for all positive integers
*n*. [a^n]^2=a^2n - Prove the inequality for the indicated integer values of
*n*. (6/5)^n, n>15 - Prove the inequality for the indicated integer values of
*n*. (7/5)^n, n>5

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### 1 Answer

A. The SuperBall :

(a) Measuring the height to which the ball rebounds after each bounce is an example of a geometric sequence. Note that the height of each successive bounce is found by **multiplying** the previous height by .9. An arithmetic sequence **adds** the same amount each time.

The height h of the ball after n bounces with an initial height of a is `h=a(.9)^n`

(b) If the ball was dropped from 2m, find the distance traveled when the ball hits the ground for the tenth time:

We could use the formula for the sum of a geometric series: `S_n=a(1-r^n)/(1-r) ` ` `where `S_n ` ` `is the sum of the first n terms, a is the first term, n is the number of terms, and r is the common ratio.

Here we get `S_n=2(1-.9^(10))/(1-.9) ` ` `. But this only accounts for the distance the ball falls each time, neglecting the upward distance. The ball goes up the same distance as the fall down so we can multiply by 2 to get:26.0528624. However, this counts an "up" of 2m; since the ball was released from 2m (it didn't bounce up to that height) we subtract 2.

**The ball bounces** 24.0528624 or **24.05m**.

B. A loan of $5000 is made with monthly repayments of $250 plus 1% of the remaining balance.

(a) This represents an arithmetic series. The balance is reduced by $250 each month, and the interest on the balance forms an arithmetic sequence with common difference $2.50.

(b) payment interest total payment new balance

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1 250 50 300 4750

2 250 47.5 297.50 4500

3 250 45 295 4250

4 250 42.5 292.5 4000

5 250 40 290 3750

6 250 37.5 287.5 3500

7 250 35 285 3250

8 250 32.5 282.5 3000

9 250 30 280 2750

10 250 27.5 277.5 2500

11 250 25 275 2250

12 250 22.5 272.5 2000

(c) The total amount of interest paid over the life of the loan -- there will be 20 payments. The first term is 50 and the common difference is 2.5. Use `S_n=n(a_1+a_n)/2 `

Here n=20,`a_1=50,a_20=2.5 ` so the total interest payments are $525